The non-F-rational locus of Rees algebras

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. In this note, we give a description of the parameter test submodule of Rees algebras. is, in turn, describes the non--rational locus. . I Let ( , ) be a -dimensional excellent local domain of prime characteristic > , where ≥ . Let be an -primary ideal. Write R ( ) for the Rees algebra ⊕ ∈N where is a variable of degree . For a domain , we write for its normalization. In this paper, we prove the following theorem: eorem . . Let ( , ) be a Cohen-Macaulay complete normal local domain of dimension at least and of characteristic > . Let be an -ideal that has a reduction generated by a system of parameters. Write R = R ( ). Suppose that R is Cohen-Macaulay. en the parameter test submodule ( R ) equals ≥ ( , ). Consequently, the non--rational locus of R is the support of the R-module R /⊕ ≥ ( , ). A reduction of is an -ideal ⊆ such that R ( ) is a finite algebra over the subring R ( ). If / is infinite, then has a reduction generated by a system of parameters. Note also that R ( ) is a finite R ( )-algebra since is excellent. Corollary . . Let ( , ) be a two-dimensional -finite Gorenstein complete local domain with an infinite residue field. Let be an -primary ideal such that := ⊕ ∈N / + (i.e. the associated graded ring for the integral closure filtration) is Gorenstein with -invariant ( ) = − . Suppose that or Proj R is -rational. en R is -rational. Our motivation for eorem . is the following result in characteristic zero. While we believe that this might have been known, we could not find a proof; hence we have included a proof in Section . We denote by J ( , ) the multiplier submodule of ; this will be defined in Section . eorem . . Let ( , ) be a Cohen-Macaulay complete normal local domain of dimension ≥ and essentially of finite type over a field of characteristic zero. Let be an -ideal such that R ( ) is Cohen-Macaulay. en the irrational locus of R ( ) is the support of the R ( )-module R /⊕ ≥ J ( , ). In Section we recall the relevant definitions and results. Section contains the proofs of eorem . , Corollary . and an example. As mentioned above, we sketch a proof of eorem . in Section .